\magnification\magstep1
\documentstyle{amsppt}
\hsize=32pc
\hoffset=4pt
\parindent=20pt
\leftheadtext{}
\rightheadtext{}
\TagsOnRight
\define\eq{\thetag} % n#mero de ecuaci"n en texto
\define\R{{\Bbb R}} % los n#meros reales
\define\V{{\Cal V}} % espacio de velocidades
\define\>#1{{\bold#1}} % notaci"n para vectores
\define\a{\alpha} % abreviatura para \alpha
\define\const{\operatorname{const}} % la funcion constante
\define\eqac{\thetag} % n#mero de ecuaci"n en texto
\define\g{{\Cal G}} % Algebra de Lie
\define\G{{\Cal G}} % Algebra de Lie
\define\mitad{\tfrac12} % la fracci"n 1/2
\define\pd#1#2{\frac{\partial#1}{\partial#2}}
% una derivada parcial
\define\set#1{\{\,#1\,\}} % notaci"n para conjuntos
\define\w{\omega} % una forma simplctica
\define\<#1>{\langle#1\rangle} % una forma bilineal
\define\X{{{\goth X}}} % un campo vectorial
\define\pr#1{{\operatorname{pr}}^{(#1)}} % para prolongaciones
\define\Sec{\operatorname{Sec}} % para Secciones
\define\Ker{\operatorname{Ker}} % para N#cleo
\define\Imagen{\operatorname{Im}} % la imagen
\define\id{\operatorname{id}} % la identidad
\define\Cin{C ^{\infty}} % conjunto de funciones diff.
\define\pic{\bigwedge} % set of unoformas
\define\A{{\Cal A}}
\define\Id{\operatorname{Id}}
\define\Lie#1{\Cal L_{{#1}}}
\define\LXa{\Cal L_{X_a}}
\define\LXb{\Cal L_{X_b}}
\define\onda{\tilde}
\define\sumaciclica{\sum_{\text{cyclic}}}
\define\x{\times}
\define\N{{\Bbb N}}
\define\End{\operatorname{End}}
\define\Poinc{\Cal P}
\define\Sim{\operatorname{Sim}}
\define\Conf{\operatorname{Conf}}
\define\conf{\operatorname{conf}}
\printoptions
\topmatter
\title
On the Relation between Weak and Strong Invariance of Differential Equations
\endtitle
\author
Jos\'e F. Cari\~nena$^\dag$, M. A. del Olmo$^\star$ and
P. Winternitz$^\ddag$
%\footnote
%{Permanent address: Centre de Recherches Mathematiques,
%Universit\'e de Montr\'eal, C.P. 6128-A, Montr\'eal (QC), H3C 3J7,
%Canada\hfill\hfill }
\endauthor
\affil
$^\dag$Departamento~de F\'{\i}sica Te\'orica,
Universidad de Zaragoza,\\
50009 Zaragoza {\smc(Spain)} \\
$^\star$Departamento~de F\'{\i}sica Te\'orica, Universidad de Valladolid, \\
47011 Valladolid {\smc(Spain)}\\
$^\ddag$Centre de Recherches Mathematiques,
Universit\'e de Montr\'eal, \\
C.P. 6128-A, Montr\'eal (QC), H3C 3J7,
{\smc(Canada).}
\endaffil
\abstract
{Some concepts of Lie algebra cohomology are used to systematize the
search for
differential equations invariant under a given Lie group $G$. In particular
it is shown that if a \lq\lq strongly invariant'' equation exists, then all
\lq\lq weakly invariant'' equations differ from it only by an arbitrary
multiplicative
factor. If no \lq\lq strongly invariant'' equation exists, then cohomology
theory can be used to
simplify the search for \lq\lq weakly invariant'' equations.}
\endabstract
\endtopmatter
\document
\head
1. Introduction.
\endhead
Lie group theory provides powerful tools for the study of differential
equations (see e.g. \cite{{1-5}} and references therein). In particular,
for nonlinear
partial differential equations, the use of Lie point symmetries makes it
possible to obtain exact analytical
solutions that are often otherwise inaccessible. The increased importance
of nonlinear phenomena in physics and other sciences,
together with developments
in computer science (algebraic computing) and in Lie group theory itself
(e.g. the theory of infinite dimensional Lie groups) have
motivated a resurgence of interest in the symmetries of differential
equations.
Lie group theory can also be used to classify differential equations
into equivalence classes, or to construct equations with predetermined
symmetries. The simplest method for doing this
consists of three steps:
\item {1.} Postulate the general form of the
invariant equation that we are looking for, i.e. the number of dependent
and independent variables and the order $n$ of the equation:
$$\Delta (x, u, u_{x_i}, u_{x_ix_j},\ldots,u_{x_{i_1}\cdots x_{i_n}})=0,
\ x\in \R^p, u\in \R^q.\tag 1.1
$$
\item {2.} Realize the Lie algebra $\G$ of the considered Lie group $G$ of
local point transformations by vector fields of the form
$$X_a=\sum_{i=1}^p\xi_i^a(x,u)\,\pd {}{x_i}+ \sum_{\a=1}^q\phi_\a^a(x,u)\,
\pd {}{u_\a}, \ a\in \G, \tag 1.2
$$
and construct the $n$-th prolongation \cite {1-5},
$\widehat X_a=\pr n X_a$, of these vector fields. Here the set
$\{X_a\}$ are the fundamental vector fields corresponding to
the elements $a$ of the Lie algebra $\G$.
\item{3.} Request that the prolongation $\widehat X_a$ of these vector
fields $X_a$ should simultaneously annihilate the function
$\Delta(x,u,u_{x_i},\ldots)$:
$$\widehat X_a\Delta =0. \tag 1.3$$
Thus, $ \Delta$ viewed as a function of $x,u$ and
the derivatives of $u$ up to order $n$, must satisfy the system of
linear first order partial differential equations (1.3). Using the method of
characteristics one finds a system of elementary solutions (if they exist) and
$\Delta$
is then an arbitrary function of these elementary invariants.
This approach has recently been applied to find equations invariant under the
symmetry
group of the Kadomtsev-Petviashvili equation \cite {6},
the Poincar\'e, Similitude and Conformal groups of $1+1$ dimensional
Minkowski space \cite {7}, and under the Galilei, Galilei-Similitude and
Schr\"odinger group of of $1+1$ dimensional space-time \cite{8}.
The approach described above has its limitations. Indeed, equation (1.3)
corresponds to the fact that the function $\Delta $ is invariant under
the corresponding local Lie point transformations. We shall call this
\lq\lq strong invariance'' of equation (1.1). Equation (1.1) is constructed out
of
\lq\lq universal invariants'' of the action of the group $G$.
What is usually required is invariance of the solution set of the
equation.
The prolongations $\widehat X_a$ thus need not annihilate $\Delta$
universally, but only on the solution set of
the equation (1.1), i.e. equation (1.3) is replaced by the weaker condition
$$\widehat X_a\Delta _{\vert \Delta =0 } =0. \tag 1.4$$
To study the relation between eq. (1.3) and (1.4), which we shall
call \lq\lq weak invariance", we note that the function $\Delta $ is not
uniquely defined. Thus, e.g.
$$
\Delta=0, \ \Delta^n=0, \ (n>0), \ e^f\Delta=0, \tag 1.5
$$
will have the same solution sets. We shall require that the equation (1.1)
define a manifold. The function $\Delta$ must be such that its
differential
does not vanish on the solution set
$$
(d\Delta)_{\vert \Delta=0}\not =0. \tag 1.6
$$
For instance $\Delta=u_{xx}$ satisfies (1.6), but $\Delta
'=\Delta^2=u_{xx}^2$ does not.
If condition (1.6) is satisfied, then the condition (1.4) of \lq\lq weak
invariance"
is (at least locally) equivalent to
$$\widehat X_a\Delta =F_a\,\Delta, \tag 1.7$$
where the functions $ F_a(x,u,u_{x_i},\ldots)$ are some, {\sl a priori}
unspecified,
functions of the same arguments as $\Delta$ that are not singular for $
\Delta=0$. We shall call equation (1.7) \lq\lq weak invariance''.
The purpose of this article is to show that if equation
(1.3) has a nontrivial
solution,
then all solutions of equation (1.4) can be obtained directly from this solution.
More specifically, if $\widetilde \Delta$ is the general solution of
equation
(1.3), and is not identically constant, then the general
solution $\Delta$ of equation (1.7)
will have the form
$$\Delta=e^f\,\widetilde\Delta, \tag 1.8$$
where $f$ is an arbitrary smooth function of the same arguments as
$\widetilde\Delta$, not singular on the solution set of the equation
$\widetilde\Delta =0$.
If, on the other hand, the Lie group action is such that no strongly
invariant equation (1.1) exists, weakly invariant equations may still be constructed.
In order to prove the essential equivalence of the set of all weakly and
strongly invariant equations (when both exist), we shall make use of
Lie algebra cohomology theory. Moreover, Lie algebra cohomology theory
can be used to simplify the task of finding weakly invariant equations
in the case when no strongly invariant ones exist.
\head
2. Lie Algebra Cohomology
\endhead
\smallskip
Let $\g$ be a Lie algebra and $\A$ a $\g$-module respectively. In other
words, $\A$ is a module that is the carrier space for a linear
representation $\Psi$ of $\g$, i.e. $\Psi \colon \g \to \End \A$ satisfies
$$\Psi (a) \Psi (b)-\Psi (b) \Psi(a)=\Psi ([a,b]).$$
By a $n$-cochain we mean an $n$-linear alternating
mapping from $\g\x\dots\x\g$ ($n$~times) into $\A$. We denote by
$C^n(\g,\A)$
the space of $n$-cochains. For every $n\in\N$ we define
$\delta_n\:C^n(\g,\A)\to
C^{n+1}(\g,\A)$ by \cite{9,10}
$$\aligned
(\delta_n\alpha)(a_1,\dots,a_{n+1})
:= &\sum_{i=1}^{n+1} (-1)^{i+1} \Psi(a_i)
\alpha(a_1,\dots,\widehat a_i,\dots,a_{n+1})+ \\
&+ \sum_{i